On the use of Set Membership theory for global optimization of black-box functions

Many science and engineering applications feature non-convex optimization problems where the performance is not explicitly modeled by a cost or reward function, i.e. it is a black box. Examples include most complex design problems where experimental tests are the main method to evaluate performance of chosen values of the decision variables, in fields such as mechanics, fluid-dynamics, electromagnetics and/or magnetohydrodynamics. Solving these problems can be done iteratively: the next value of the decision variables is chosen based on the outcome generated by the previous tests. The time and resource overhead in conducting tests, however, raises the issue of most efficiently choosing the next test point according to previous observations. To approach this issue, a new global optimization strategy based on a Set Membership framework is proposed. Assuming a Lipschitz continuous cost function, the presented algorithm builds an approximation of the latter to decide whether to exploit the best result obtained so far, or to further explore the decision space. The proposed algorithm is presented and some implementation aspects are discussed. Its performance is evaluated on a set of benchmark non-convex problems and compared with those of other global optimization approaches.